You're not completely off the mark with BLM, either. But it's more in their Lore rather than their gameplay. Basically from the BLM quests the Mhachi were indeed big on voidsents, but the remnants of the art of Black Magic focus more on using ambient aether to power destructive spells, in lore this even meant summoning frigging meteors on the land (which Shattotto did, and which BLM does in their LB3). Modern "Black Magic" is mixed with Thaumaturgy, which was created for ritualistic purposes in Ul'dah to preserve and purify bodies in funeral rites.

In gameplay, it also comes from (as Vyrerus said) traditional FF Black Magic, which is focused on elemental magic with only SOME unaspected but "dark" spells (like Meteor, or Death). WHAT elements are available varies from FF to FF, with Aero sometimes being Black Magic and sometimes being White Magic; or Water being available in some FFs and unavailable in others. But the standard BLM spell list will feature Fire, Ice, and Lightning.

In broad strokes, everything every class does is a manipulation of aether, and the only difference is HOW they manipulate said aether: even MNK is stated that their chakra techniques are basically aether manipulation using their own bodies. PLD/GLD USED to let out a burst of aether as a bright light with Flash, but that's been removed...